Both sentences are fine. "The will" or "his will" is correct.
Student or Learner
Would you please take a look at my sentences and correct my mistakes?
1.The children accused their father of making the will when he was intoxicated. ( do I need to write "the will or his will"?)
2. A well-known painter willed his young lover a large house, which caused his family to rage.
In the wider context of the text in which this sentence occurred, the will could already have been mentioned. In that case, I would do the same as Rover_KE.